Weyl Symmetry, Planck Scale, & Inflation.

1. Weyl Symmetry, Planck Scale, & Inflation. Christopher T. Hill Fermilab Eclipse Conference Celebration of Thom Curtright August, 2017

2. Inflation and Planck Scale Generation as a Unified Phenomenon No fifth force in a scale invariant universeBy Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross.arXiv:1612.03157 [gr-qc], to appearTPhys. Rev .D.Weyl Current, Scale-Invariant Inflation and Planck Scale GenerationBy Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross.arXiv:1610.09243 [hep-th]. Phys.Rev. D95 (2017) no.4, 043507.Scale-Independent Inflation and Hierarchy GenerationBy Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross.arXiv:1603.05983 [hep-th].Phys.Lett. B763 (2016) 174-178.

3. Related earlier works …..

4. Weyl Invariance in a nutshell: Coordinates are scale free numbers. Length is defined by the covariant metric. Fields have canonical mass (length)-1 dimension Local Weyl transformation: Global Weyl transformation:

5. Some Local Weyl Invariants:

6. Exercise: Construct a locally Weyl invariant action Integrate by parts:

7. A globally Weyl = Scale invariant action in D=4:

8. A globally Weyl = Scale invariant action in D=4: Define:

9. A globally Weyl = Scale invariant action in D=4: Define: Total divergence Canonical normalization of s: f

10. Result: ``Jordan frame” ``Einstein frame” Weyl symmetry in Jordan frame is hidden in the Einstein frame. Can show that s decouples from Fermions and gauge fields. (No Brans-Dicke constraints.)

11. Note singularity: a - 1 = 0 (local Weyl invariance) s terms cancel Note wrong sign when a = 1

12. A conventional scalar-gravity action: (notation g = (1,-1,-1,-1))

13. A conventional scalar-gravity action: (notation g = (1,-1,-1,-1)) A scale invariant scalar-gravity action:

14. A conventional scalar-gravity action: (notation g = (1,-1,-1,-1)) A scale invariant scalar-gravity action: These theories are Classically Weyl Equivalent

15. A conventional scalar-gravity action: (notation g = (1,-1,-1,-1)) A scale invariant scalar-gravity action: How do we understand the dynamics of the first theory (Einstein frame) in the second theory (Jordan frame)?

16. Compute in the Jordan Frame (!)!!! Einstein Equations Compare:

17. Trace of Einstein Equations

18. Klein-Gordon Equation

19. Trace and KG Equations: Combine to eliminate R:

20. Trace and KG Equations Combine to eliminate R:

21. Combined Trace of Einstein and KG Equations yields a current conservation law: If then conserved Conserved Current :

22. The Kernel of the Current The current can be written as where is the “kernel”

23. The Kernel of the Current The current can be written as where is the “kernel” FRW: becomes

24. The Kernel of the Current constant constant “Inertial” spontaneous scale generation

25. The Kernel of the current Spatially constant Einstein equations Theory leads to eternal inflation

26. The Kernel of the current constant

27. Our action is globally Weyl invariant: Noether: Perform a local infinitesimal Weyl Transformation:

28. Our action is globally Weyl invariant: Noether Current: properties: constant

29. Inertial spontaneous breaking of global Weyl symmetry leads to dilaton: Define:

30. A conventional scalar-gravity action: A scale free action Summary: These theories are classically equivalent; dynamical equivalence follows from Weyl current.

31. Can our world have a hidden Weyl invariance? How can we inflate and exit eternal inflation?

32. Two Scalars, globally Weyl symmetric Effective Planck Mass

33. Two Scalar K-current Two Scalar Potential Current divergence Kernel

34. Potential with a flat direction:

35. Fixed K defines an ellipse

36. Fixed K defines an ellipse

39. below. More detailed studies in progress (Ferreira) Scale-Independent Inflation and Hierarchy GenerationBy Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross.arXiv:1603.05983 [hep-th].Phys.Lett. B763 (2016) 174-178. Other phenomena? Dilaton stars and clumps? (maybe seed primordial black holes) Dilatonic hair? (hard) Dilatonic radiation from BH annihilation? (final burst)

40. For this to work Weyl symmetry must be exact: (Failure is not an option) Does this conflict with Quantum Mechanics?

41. Scale = Weyl Symmetry appears ab initio to be problematic due to loop divergences. However, loop divergences are subtle and are often confused with physics.

42. An Operating Principle (W. Bardeen) Quantum loop divergences are unphysical; they are artifacts of the method of calculation. The Allowed Symmetries of a Renormalized Quantum Field Theory are determined by anomalies, or absence thereof. There are no anomalies associated with quadratic or quartic divergences; Scale Symmetry is permitted modulo Trace Anomalies; trace anomalies are D=4 ops.

43. A Common Complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry !!! What are these people talking about? “

44. A Common Complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry !!! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “

45. These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry ! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ It is False to conclude that the theory does not exist because the regulator doesn’t respect the defining symmetry.

46. These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry ! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ The regulator one uses may break symmetries. For gauge theories consistent regulators exist that moderate quadratic divergences, eg, Pauli-Villars, Dim. reg.

47. These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry ! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ No regulator unambiguously deals with chiral anomalies !!! These always require human intervention (ie, counterterms to define loops)

48. These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ For scale symmetry we do not have a consistent regulator. The physics cannot depend upon the choice of regulator! Scale symmetry can be maintained at loop level if trace anomaly cancels.

49. “The Higgs Boson Mass receives a contribution coming from the cutoff used to compute loops”

50. “The Higgs Boson Mass receives a contribution coming from the cutoff used to compute loops” FALSE. However the Higgs can mix with GUT or Gravity sectors, and this is a real physics problem that must be controlled.